I am looking at running an FT232RL as well.
Use a CY7C65213 instead, better then the FTDI stuff
What is confusing me at the moment is it has been hard to find information regarding non-arduino based AVR bootloaders or AVR bootloaders in general.
(...)
that the Arduino bootloader for the Arduino Mega 2560 should work as a bootloader? I don't want to assume though.
You either upload your program with a dedicated programmer through the ATMega2560 programing interface or use a bootloader to upload your code though the serial interface.
But you have to burn the bootloader at least once with the "real" programmer.
Try a google search with "AVRISP MK2", "AVR Dragon" or "USBASP".
Those are programmers.
For a search yielding results on the bootloader, ty "stk500 v2 atmega2560".
The programmer is a tad faster then the bootloader, uloading 200kb of code through the bootloader will take a decent amount of time.
Currently i have a ATMega2560 on a breadboard, been doing something similar as you.
Alltough it is not a calculator but it could be... ist a matter of the firmware...